Well, they specifically chose an 'Honoured Britannian' like Suzaku to frame for the death of the governor because as the 'Purist Faction', they want only Britannians as a part of Britannia. As such, this was a perfect opportunity to get rid of the Elevens that belonged to Britannia as part of the Honoured Britannian system. As for why Suzaku in particular... it might've been the fact that he was the only survivor from the group that went to find the poison gas container (it seems like all the others were killed by Lelouch), or most likely it was a plot device that allowed the writer to introduce Lulu's "Zero" persona.

At this point, pretty much. Though I think her name gets dropped in episode 5 or 6...

Suzaku was also the son of japan's last prime minister. One of the main reasons why he chosen.